La journée 2026 de l’équipe Combinatoire et Optimisation de l’IMJ-PRG a eu lieu le 8 septembre. Voici le programme de la journée :
14h00-14h20 Giulia Menara: « A Helly-type theorem for two-component convex sets »
Abstract: In any fixed dimension, consider a finite collection of sets, each consisting of exactly two disjoint, closed, convex pieces. We show that to guarantee the intersection of the whole collection also consists of exactly two such convex pieces, it suffices to verify the same two-piece structure for all intersections of some subfamilies of intermediate size, thus answering a question of Gil Kalai.
14h20-14h40 Maena Quemener: « A New Model for Modified Hall–Littlewood Polynomial Coefficients »
Abstract: Recently, Kim, Lee, and Yoo gave a Hall–Littlewood expansion of the chromatic quasisymmetric function and described its coefficients using a new combinatorial model: linked rook placements (LRPs). We give a new formula for the statistic on LRPs and construct a bijection between LRPs and P-arrays. This allows us to give a new combinatorial description of the coefficients of modified Hall–Littlewood polynomials in terms of linked rook placements.
14h40-15h00 Joel Hakavuori: « Macaulay defects and a conjecture of Kalai »
Abstract: By the g-theorem, the g-vector of a simplicial polytope P, which encodes the numbers of faces of each dimension of P, is the Hilbert function of an Artinian quotient of a polynomial ring. This implies that the entries of the g-vector are constrained by Macaulay’s growth conditions. Kalai conjectured that for a sequence of simplicial polytopes approximating a C^1 convex body, the entries of the g-vector as well as the defects to extremal growth must diverge. Adiprasito, Nevo, and Samper proved the first assertion, and we build on their construction of many local affine stresses to give a proof of the second statement.
15h00-15h20 Jules Tsukahara: « Newton Polygons and Learning Coefficients in Two Dimensions »
Abstract: Local learning coefficients measure the effective dimension of a machine learning model’s loss landscape, and have recently gained traction in AI interpretability. Remarkably, they coincide with a classical invariant from algebraic geometry — the local real log canonical threshold of the loss — yet general methods for computing them exactly remain limited. After explaining this connection between algebraic geometry and statistical learning theory, we show how to compute local learning coefficients for two-parameter models whose loss is contact-equivalent to a polynomial. The key turns out to be combinatorial: the answer can be read off a Newton polygon.
15h20-15h50 pause café
15h50-16h10 Sofia Zotova: « The Erdős-Ginzburg-Ziv constant of rank-two-like p-groups »
Abstract: The Davenport constant of a finite abelian group G is defined as the smallest positive integer t such that every sequence of at least t elements of G contains a nonempty subsequence with sum zero. The Erdős-Ginzburg-Ziv constant of G is defined as the smallest positive integer t such that every sequence of at least t elements of G contains a subsequence with sum zero and length equal to the exponent of G. We have improved a theorem of Schmid and Zhuang bounding the Erdős-Ginzburg-Ziv constant of p-groups, the Davenport constant of which is at most twice their exponent. Such groups are called rank-two-like groups. In particular, this gives the exact value of the Erdős-Ginzburg-Ziv constant, confirming a conjecture of Gao, for certain rank-two-like p-groups. Joint work with Benjamin Girard.
16h10-16h30 Vasiliki Petrotou: « Parseval-Rayleigh identities for Cohen-Macaulay rings »
Abstract: Parseval–Rayleigh identities, originally arising in Fourier analysis, have recently acquired an algebraic counterpart in positive characteristic that has proven effective in the study of Lefschetz properties of graded algebras associated with combinatorial objects. As a consequence, these identities have played a central role in resolving unimodality conjectures in combinatorics. In this talk, we study these identities from the perspective of commutative algebra and explain their origin. As a special case of the general framework, we analyze the Parseval-Rayleigh identities for homogeneous complete intersections focusing on their connection to Lefschetz properties for this class. This is joint work with K. Adiprasito, E. Katz, R. Oba and S. Papadakis.
16h30-16h50 Ryoshun Oba: « Lefschetz property for F-pure rings »
Abstract: I will talk about an application of the Parseval-Rayleigh identity to the Lefschetz property for F-pure rings.